Factorization solution of a family of quantum nonlinear oscillators

In a recent paper, Carinena J F, Perelomov A M, Ranada M F and Santander M (2008 J. Phys. A: Math. Gen. 41 085301) analyzed a non-polynomial one-dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schrodinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can exactly be solved in a similar manner.